[proofplan] We prove the colimit universal property for the cocone with vertex $F(C)$. For each object $d \in \mathcal D$, a morphism $F(C) \to d$ determines a cocone from $F \circ D$ to $d$ by precomposition with the morphisms $F(\iota_j)$. The adjunction identifies such cocones with cocones from $D$ to $G(d)$, and the colimit property of $C$ identifies those with morphisms $C \to G(d)$. Applying the adjunction once more gives a unique morphism $F(C) \to d$, which is exactly the desired universal property. [/proofplan] [step:Fix the adjunction bijections and the candidate cocone] Let $(C,(\iota_j:D(j)\to C)_{j \in \operatorname{Ob}(J)})$ denote the given colimit cocone for the diagram $D:J\to \mathcal C$. Let \begin{align*} \Phi_{a,d}: \operatorname{Hom}_{\mathcal D}(F(a),d) &\longrightarrow \operatorname{Hom}_{\mathcal C}(a,G(d)) \end{align*} denote the adjunction bijection, defined for every object $a \in \mathcal C$ and every object $d \in \mathcal D$. The naturality identity used below is that for every morphism $u: a' \to a$ in $\mathcal C$ and every morphism $v: F(a) \to d$ in $\mathcal D$, \begin{align*} \Phi_{a',d}(v \circ F(u)) = \Phi_{a,d}(v) \circ u. \end{align*} For every morphism $\alpha: j \to k$ in $J$, the original cocone satisfies \begin{align*} \iota_k \circ D(\alpha)=\iota_j. \end{align*} Applying the functor $F$ gives \begin{align*} F(\iota_k) \circ F(D(\alpha)) = F(\iota_j), \end{align*} so $(F(\iota_j):F(D(j))\to F(C))_{j \in \operatorname{Ob}(J)}$ is a cocone from $F \circ D$ to $F(C)$ in $\mathcal D$. [/step] [step:Translate cocones under the adjunction] Fix an object $d \in \mathcal D$. A cocone from $F \circ D$ to $d$ is a family \begin{align*} (\beta_j:F(D(j))\to d)_{j \in \operatorname{Ob}(J)} \end{align*} such that for every morphism $\alpha:j\to k$ in $J$, \begin{align*} \beta_k \circ F(D(\alpha))=\beta_j. \end{align*} Define a family $(\gamma_j:D(j)\to G(d))_{j \in \operatorname{Ob}(J)}$ by \begin{align*} \gamma_j := \Phi_{D(j),d}(\beta_j). \end{align*} For every morphism $\alpha:j\to k$ in $J$, naturality of $\Phi$ in the $\mathcal C$-variable gives \begin{align*} \gamma_k \circ D(\alpha) &= \Phi_{D(k),d}(\beta_k)\circ D(\alpha) \\ &= \Phi_{D(j),d}(\beta_k\circ F(D(\alpha))) \\ &= \Phi_{D(j),d}(\beta_j) \\ &= \gamma_j. \end{align*} Thus $(\gamma_j)_{j \in \operatorname{Ob}(J)}$ is a cocone from $D$ to $G(d)$. Conversely, if $(\gamma_j:D(j)\to G(d))_{j \in \operatorname{Ob}(J)}$ is a cocone from $D$ to $G(d)$, define \begin{align*} \beta_j := \Phi_{D(j),d}^{-1}(\gamma_j). \end{align*} Let $\alpha:j\to k$ be a morphism in $J$. Naturality of $\Phi$ in the $\mathcal C$-variable gives \begin{align*} \Phi_{D(j),d}(\beta_k\circ F(D(\alpha))) &= \Phi_{D(k),d}(\beta_k)\circ D(\alpha) \\ &= \gamma_k\circ D(\alpha) \\ &= \gamma_j \\ &= \Phi_{D(j),d}(\beta_j). \end{align*} Since $\Phi_{D(j),d}$ is injective, it follows that \begin{align*} \beta_k \circ F(D(\alpha))=\beta_j. \end{align*} Hence the adjunction bijections identify cocones from $F\circ D$ to $d$ with cocones from $D$ to $G(d)$. [/step] [step:Use the colimit property of $C$ to obtain the unique mediating morphism] Let $(\beta_j:F(D(j))\to d)_{j \in \operatorname{Ob}(J)}$ be a cocone from $F\circ D$ to $d$. By the previous step, the family \begin{align*} \gamma_j := \Phi_{D(j),d}(\beta_j) \end{align*} is a cocone from $D$ to $G(d)$. Since $(C,(\iota_j)_{j \in \operatorname{Ob}(J)})$ is a colimit cocone for $D$, there exists a unique morphism \begin{align*} h:C\to G(d) \end{align*} in $\mathcal C$ such that for every object $j \in \operatorname{Ob}(J)$, \begin{align*} h\circ \iota_j=\gamma_j. \end{align*} Define \begin{align*} \bar h := \Phi_{C,d}^{-1}(h):F(C)\to d. \end{align*} For each object $j \in \operatorname{Ob}(J)$, naturality of the adjunction bijection gives \begin{align*} \Phi_{D(j),d}(\bar h\circ F(\iota_j)) &= \Phi_{C,d}(\bar h)\circ \iota_j \\ &= h\circ \iota_j \\ &= \gamma_j \\ &= \Phi_{D(j),d}(\beta_j). \end{align*} Since $\Phi_{D(j),d}$ is injective, it follows that \begin{align*} \bar h\circ F(\iota_j)=\beta_j. \end{align*} Thus $\bar h:F(C)\to d$ mediates the cocone $(\beta_j)_{j \in \operatorname{Ob}(J)}$ through the cocone $(F(\iota_j))_{j \in \operatorname{Ob}(J)}$. [/step] [step:Prove uniqueness of the mediating morphism] Suppose $q:F(C)\to d$ is another morphism in $\mathcal D$ such that \begin{align*} q\circ F(\iota_j)=\beta_j \end{align*} for every object $j \in \operatorname{Ob}(J)$. Define \begin{align*} r:=\Phi_{C,d}(q):C\to G(d). \end{align*} For every object $j \in \operatorname{Ob}(J)$, naturality of the adjunction bijection gives \begin{align*} r\circ \iota_j &= \Phi_{C,d}(q)\circ \iota_j \\ &= \Phi_{D(j),d}(q\circ F(\iota_j)) \\ &= \Phi_{D(j),d}(\beta_j) \\ &= \gamma_j. \end{align*} By uniqueness in the colimit property of $C$, we have $r=h$. Since $\Phi_{C,d}$ is injective, $q=\bar h$. Therefore, for every object $d \in \mathcal D$ and every cocone from $F\circ D$ to $d$, there exists a unique morphism $F(C)\to d$ through which that cocone factors. This is precisely the universal property of the colimit of $F\circ D$. Hence $F(C)$, equipped with the cocone $(F(\iota_j))_{j \in \operatorname{Ob}(J)}$, is a colimit of $F\circ D$ in $\mathcal D$. [/step]