[proofplan] We construct the limit algebra on the set-theoretic limit of the underlying diagram. Each operation is defined coordinatewise, and compatibility of tuples follows because every morphism in the diagram preserves the operations. The equations hold because they hold in each coordinate. Finally, the universal property is inherited from the set-theoretic limit, and the induced map from any cone is a homomorphism because all its coordinates are homomorphisms. [/proofplan]

[step:Build the underlying set-theoretic limit] Let $J$ be a small category, and let $D:J\to\mathcal{V}$ be a diagram. For each object $j\in\operatorname{Ob}(J)$, write $D(j)$ for the corresponding $\Sigma$-algebra, and for each morphism $f:j\to k$ in $J$, write \begin{align*} D(f):D(j)\to D(k) \end{align*} for the corresponding $\mathcal{V}$-homomorphism. Let $L$ be the set-theoretic limit of the underlying diagram $UD:J\to\mathsf{Set}$. Explicitly, \begin{align*} L = \left\{ (x_j)_{j\in\operatorname{Ob}(J)}\in \prod_{j\in\operatorname{Ob}(J)} UD(j) : UD(f)(x_j)=x_k \text{ for every } f:j\to k \right\}. \end{align*} For each $j\in\operatorname{Ob}(J)$, define the projection map \begin{align*} \pi_j:L&\to UD(j),\\ (x_i)_{i\in\operatorname{Ob}(J)}&\mapsto x_j. \end{align*} The family $(\pi_j)_{j\in\operatorname{Ob}(J)}$ is the limiting cone of $UD$ in $\mathsf{Set}$. [/step]

[step:Define every algebraic operation coordinatewise] Let $\omega\in\Sigma$ be an operation symbol of arity $n\in\mathbb{N}\cup\{0\}$. For $n>0$, define \begin{align*} \omega_L:L^n&\to L,\\ (a_1,\dots,a_n)&\mapsto \left( \omega_{D(j)}\bigl(\pi_j(a_1),\dots,\pi_j(a_n)\bigr) \right)_{j\in\operatorname{Ob}(J)}. \end{align*} For $n=0$, define $\omega_L\in L$ by \begin{align*} \pi_j(\omega_L)=\omega_{D(j)} \end{align*} for every $j\in\operatorname{Ob}(J)$. We verify that these definitions actually land in $L$. Let $f:j\to k$ be a morphism of $J$. Since $D(f):D(j)\to D(k)$ is a $\mathcal{V}$-homomorphism, it preserves the operation $\omega$. Thus, for $n>0$ and $a_1,\dots,a_n\in L$, \begin{align*} UD(f)\left(\omega_{D(j)}(\pi_j(a_1),\dots,\pi_j(a_n))\right) &= \omega_{D(k)}\left(UD(f)(\pi_j(a_1)),\dots,UD(f)(\pi_j(a_n))\right)\\ &= \omega_{D(k)}\left(\pi_k(a_1),\dots,\pi_k(a_n)\right), \end{align*} because each $a_i\in L$ is a compatible tuple. The nullary case is the same calculation with no inputs: \begin{align*} UD(f)(\omega_{D(j)})=\omega_{D(k)}. \end{align*} Therefore each coordinatewise operation is well-defined on $L$. [/step]

[step:Verify the defining equations coordinate by coordinate] Let $s=t$ be an equation in the defining set $E$, where $s$ and $t$ are $\Sigma$-terms in variables $x_1,\dots,x_m$. For each $\Sigma$-algebra $A$, write \begin{align*} s_A:A^m\to A, \qquad t_A:A^m\to A \end{align*} for the term operations determined by $s$ and $t$. We claim that for every term $r$ in variables $x_1,\dots,x_m$, every $a_1,\dots,a_m\in L$, and every $j\in\operatorname{Ob}(J)$, \begin{align*} \pi_j\bigl(r_L(a_1,\dots,a_m)\bigr) = r_{D(j)}\bigl(\pi_j(a_1),\dots,\pi_j(a_m)\bigr). \end{align*} This follows by induction on the construction of the term $r$: it is immediate for variables, and the induction step is exactly the coordinatewise definition of the operation symbols. Applying this formula to $s$ and $t$, and using that $D(j)$ satisfies every equation in $E$, gives \begin{align*} \pi_j\bigl(s_L(a_1,\dots,a_m)\bigr) &= s_{D(j)}\bigl(\pi_j(a_1),\dots,\pi_j(a_m)\bigr)\\ &= t_{D(j)}\bigl(\pi_j(a_1),\dots,\pi_j(a_m)\bigr)\\ &= \pi_j\bigl(t_L(a_1,\dots,a_m)\bigr) \end{align*} for every $j\in\operatorname{Ob}(J)$. Since elements of $L$ are tuples and equality in $L$ is coordinatewise equality, we have \begin{align*} s_L(a_1,\dots,a_m)=t_L(a_1,\dots,a_m). \end{align*} Thus $L$, equipped with the coordinatewise operations, is an object of $\mathcal{V}$. [/step]

[step:Show the projections are homomorphisms] For each $j\in\operatorname{Ob}(J)$, the projection \begin{align*} \pi_j:L\to D(j) \end{align*} preserves every operation by construction. Indeed, for an $n$-ary operation symbol $\omega\in\Sigma$ with $n>0$ and elements $a_1,\dots,a_n\in L$, \begin{align*} \pi_j\bigl(\omega_L(a_1,\dots,a_n)\bigr) = \omega_{D(j)}\bigl(\pi_j(a_1),\dots,\pi_j(a_n)\bigr). \end{align*} For a nullary operation symbol $\omega$, the definition gives \begin{align*} \pi_j(\omega_L)=\omega_{D(j)}. \end{align*} Therefore each $\pi_j$ is a $\mathcal{V}$-homomorphism. Since the underlying maps form a cone over $UD$, the homomorphisms $(\pi_j)_{j\in\operatorname{Ob}(J)}$ form a cone over $D$ in $\mathcal{V}$. [/step]

[step:Prove the universal property in $\mathcal{V}$] Let $A$ be an object of $\mathcal{V}$, and let \begin{align*} \alpha_j:A\to D(j) \end{align*} be a cone over $D$ in $\mathcal{V}$. By the universal property of the set-theoretic limit $L$, there is a unique function \begin{align*} h:UA\to L \end{align*} such that \begin{align*} \pi_j\circ h=U\alpha_j \end{align*} for every $j\in\operatorname{Ob}(J)$. We verify that $h$ is a $\mathcal{V}$-homomorphism. Let $\omega\in\Sigma$ be an operation symbol of arity $n>0$, and let $a_1,\dots,a_n\in UA$. For every $j\in\operatorname{Ob}(J)$, \begin{align*} \pi_j\left(h\bigl(\omega_A(a_1,\dots,a_n)\bigr)\right) &= \alpha_j\bigl(\omega_A(a_1,\dots,a_n)\bigr)\\ &= \omega_{D(j)}\bigl(\alpha_j(a_1),\dots,\alpha_j(a_n)\bigr)\\ &= \pi_j\left(\omega_L(h(a_1),\dots,h(a_n))\right). \end{align*} Since equality in $L$ is coordinatewise equality, \begin{align*} h\bigl(\omega_A(a_1,\dots,a_n)\bigr) = \omega_L(h(a_1),\dots,h(a_n)). \end{align*} For a nullary operation symbol $\omega$, the same argument gives \begin{align*} \pi_j(h(\omega_A)) = \alpha_j(\omega_A) = \omega_{D(j)} = \pi_j(\omega_L) \end{align*} for every $j$, hence $h(\omega_A)=\omega_L$. Thus $h:A\to L$ is a $\mathcal{V}$-homomorphism. If $g:A\to L$ is any $\mathcal{V}$-homomorphism satisfying $\pi_j\circ g=\alpha_j$ for every $j$, then $Ug:UA\to L$ has the same composites with the set-theoretic limiting projections as $h$. By uniqueness in $\mathsf{Set}$, $Ug=h$, so $g=h$ as a morphism in $\mathcal{V}$. Therefore $(L,(\pi_j))$ is a limit cone for $D$ in $\mathcal{V}$. [/step]

[step:Conclude that the forgetful functor creates the limit] The $\mathcal{V}$-algebra structure on the set $L$ is forced by the requirement that every projection $\pi_j:L\to D(j)$ preserve every operation: for each operation symbol $\omega\in\Sigma$, the $j$-th coordinate of $\omega_L$ must be the corresponding operation in $D(j)$. Hence the lifted algebra structure is unique. We have shown that the set-theoretic limit cone of $UD$ lifts uniquely to a cone in $\mathcal{V}$ and that the lifted cone is limiting in $\mathcal{V}$. Since $J$ and $D$ were arbitrary, the forgetful functor \begin{align*} U:\mathcal{V}\to\mathsf{Set} \end{align*} creates all small limits. [/step]