It suffices to show this for psuedo-colimits, i.e. ones for which the induced equivalence of categories is in fact an isomorphism of categories (such a strict model for the weak colimit exists, since we can explicitly calculate one in the 2-category of groupoids). Denote pseudo-colimits by $pcolim$.

Some notation:

If $\mu:F \Rightarrow \Delta_X$ is a cocone, and $f:X \to Y$, denote by $\hat{\mu}(f)$ the cocone $\Delta_{f} \circ \mu:F \Rightarrow \Delta_{Y}$.

Let $j$ denote the canonical $2$-functor from strict presheaves in groupoids into weak ones.

Let $F:J \to Psh\left(C,Gpd\right)$ Psh\left(\mathcal{C},Gpd\right)$be any pseudo-functor. Let $$\mu_{S}:F \Rightarrow \Delta_{pcolim F}$$ be a pseudo-colimiting cocone for$F$and $$\mu_{W}:j \circ F \Rightarrow \Delta_{pcolim j \circ F}$$ be a pseudo-colimiting cocone for$j \circ F$. To simplify notation, let$S:=pcolim F$and$W:=pcolim j\circ F$. Then $$j\mu_{S}:j\circ F \Rightarrow \Delta{j\left(S\right)}$$ is a cocone for$j \circ F$with vertex$j\left(S\right)$. Hence there exists a morphism$\phi:W \to S$such that$j\mu_{S}=\Delta_{\phi} \circ \mu_{W}$. We claim that$\phi$is an isomorphism. It suffices to show that for each$C \in C_0$, \mathcal{C}_0$, the map $\phi\left(C\right):W\left(C\right) \to S\left(C\right)$ is an isomorphism of groupods. Consider the inclusion of the object $C$ as a functor $$1\stackrel{\imath}{\rightarrow} C$$ \mathcal{C}$$from the terminal category. This induces two 2-functors, and by abuse of notation, we will denote both by $$\imath^{*}:Psh\left(C,Gpd\right) \to Gpd$$ $$\imath^{*}:Gpd^{C^{op}} \to Gpd.$$ Both of these 2-functors are pseudo left adjoints and are given by evaluation at the objet C, so clearly \imath^{*}j=\imath^{*}. We want to show that $$\phi\left(C\right)=\imath^{*}\left(\phi\right)$$ is an isomorphism of groupoids. Since \imath^{*} is a pseudo left adjoint, it follows that $$\imath^{*}\mu_{W}:\imath^{*}\circ j \circ F=F\left(C\right) \Rightarrow \imath^{*} \circ \Delta_{W}= \Delta_{W\left(C\right)}$$ is pseudo-colimiting. Now, since pseudo-colimits Psh\left(C,Gpd\right) are computed pointwise, it follows that \imath^{*}\mu{S} is pseudo-colimiting. So, there exists a functor $$\psi:S\left(C\right) \to W\left(C\right)$$ such that $$\Delta{\psi}\circ\imath^{*}\mu_{S}=\imath^{*}\mu_{W}.$$ Notice that$$\imath^{*}\mu_{S}=\imath^{*}\left(\Delta_{\phi}\right)\circ \mu_{W})=\Delta_{\phi\left(C\right)}\circ \imath^{*}\mu_{W}.$$So$$\Delta_{\psi}\circ \Delta{\phi\left(C\right)}\circ \imath^{*}\mu_{W}=\imath^{*}\mu_{W}.$$The left hand side of this equation is equal to \hat{\imath^{*}\mu_{W}}\left(\psi \circ \phi\left(C\right)\right) whereas the right hand side is equal to \hat{\imath^{*}\mu_{W}}\left(id_{W\left(C\right)}\right). But \imath^{*}\mu_{W} is pseudo-colimiting, so \hat{\imath^{*}\mu_{W}} is an isomorphism of categories, hence \psi \circ \phi\left(C\right)=id_{W\left(C\right)}. Notice further that \begin{eqnarray*} \imath^{*}\mu_{S}\left(\phi\left(C\right) \circ \psi\right)&=&\Delta_{\phi\left(C\right)} \circ \Delta{\psi} \circ \Delta{\phi\left(C\right)} \circ \imath^{*}\mu_{W}\ &=&\Delta_{\phi\left(C\right)} \circ \imath^{*}\mu_{W}\ &=&\imath^{*}\mu_{S}\ &=&\hat{\imath^{*}\mu_{S}}\left(id_{S\left(C\right)}\right).\ \end{eqnarray*} But \imath^{*}\mu_{S} is pseudo-colimiting, so \hat{\imath^{*}\mu_{S}} is an isomorphism, hence$$\phi\left(C\right) \circ \psi=id_{S\left(C\right)}.$$Therefore, \phi is an isomorphism. Since pseudo-colimits are stable under isomorphisms, it follows that j\left(S\right) is a pseudo-colimit for j\circ F. 4 Tex fix It suffices to show this for psuedo-colimits, i.e. ones for which the induced equivalence of categories is in fact an isomorphism of categories (such a strict model for the weak colimit exists, since we can explicitly calculate one in the 2-category of groupoids). Denote pseudo-colimits by pcolim. Let j denote the canonical 2-functor from strict presheaves in groupoids into weak ones. Let F:J \to Psh\left(C,Gpd\right) be any pseudo-functor. Let$$\mu_{S}:F \Rightarrow \Delta_{pcolim F}$$be a pseudo-colimiting cocone for F and$$\mu_{W}:j \circ F \Rightarrow \Delta_{pcolim j \circ F}$$be a pseudo-colimiting cocone for j \circ F. To simplify notation, let S:=pcolim F and W:=pcolim j\circ F. Then$$j\mu_{S}:j\circ F \Rightarrow \Delta{j\left(S\right)}$$is a cocone for j \circ F with vertex j\left(S\right). Hence there exists a morphism \phi:W \to S such that j\mu_{S}=\Delta_{\phi} \circ \mu_{W}. We claim that \phi is an isomorphism. It suffices to show that for each C \in C_0, the map \phi\left(C\right):W\left(C\right) \to S\left(C\right) is an isomorphism of groupods. Consider the inclusion of the object C as a functor$$1\stackrel{\imath}{\rightarrow} C$$from the terminal category. This induces two 2-functors, and by abuse of notation, we will denote both by '$$\imath^{}:Psh\left(C,Gpd\right) $$\imath^{*}:Psh\left(C,Gpd\right) \to Gpd$$' '$$\imath^{}:Gpd^{C^{op}}  $$\imath^{*}:Gpd^{C^{op}} \to Gpd.$$'  Both of these 2-functors are pseudo left adjoints and are given by evaluation at the objet C, so clearly \imath^{}j=\imath^{}.\imath^{*}j=\imath^{*}. We want to show that '$$\phi\left(C\right)=\imath^{}\left(\phi\right)$$' $$\phi\left(C\right)=\imath^{*}\left(\phi\right)$$ is an isomorphism of groupoids. Since \imath^{}\imath^{*} is a pseudo left adjoint, it follows that '$$\imath^{*}\mu_{W}:\imath^{}\circ $$\imath^{*}\mu_{W}:\imath^{*}\circ j \circ F=F\left(C\right) \Rightarrow \imath^{} imath^{*} \circ \Delta_{W}= \Delta_{W\left(C\right)}$$' Delta_{W\left(C\right)}$$ is pseudo-colimiting. Now, since pseudo-colimits 'Psh\left(C,Gpd\right)' Psh\left(C,Gpd\right) are computed pointwise, it follows that '\imath^{}\mu{S}' \imath^{*}\mu{S} is pseudo-colimiting. So, there exists a functor '$$\psi:S\left(C\right) $$\psi:S\left(C\right) \to W\left(C\right)$$'  such that '$$\Delta{\psi}\circ\imath^{}\mu_{S}=\imath^{*}\mu_{W}.$$'$$\Delta{\psi}\circ\imath^{*}\mu_{S}=\imath^{*}\mu_{W}.$$

Notice that $$\imath^{*}\mu_{S}=\imath^{*}\left(\Delta_{\phi}\right)\circ \mu_{W})=\Delta_{\phi\left(C\right)}\circ \imath^{*}\mu_{W}.$$ So

$$\Delta_{\psi}\circ \Delta{\phi\left(C\right)}\circ \imath^{*}\mu_{W}=\imath^{*}\mu_{W}.$$ The left hand side of this equation is equal to $\hat{\imath^{*}\mu_{W}}\left(\psi \circ \phi\left(C\right)\right)$ whereas the right hand side is equal to $\hat{\imath^{*}\mu_{W}}\left(id_{W\left(C\right)}\right).$ But $\imath^{*}\mu_{W}$ is pseudo-colimiting, so $\hat{\imath^{*}\mu_{W}}$ is an isomorphism of categories, hence $\psi \circ \phi\left(C\right)=id_{W\left(C\right)}.$

Notice further that

\begin{eqnarray*} \imath^{*}\mu_{S}\left(\phi\left(C\right) \circ \psi\right)&=&\Delta_{\phi\left(C\right)} \circ \Delta{\psi} \circ \Delta{\phi\left(C\right)} \circ \imath^{*}\mu_{W}\ &=&\Delta_{\phi\left(C\right)} \circ \imath^{*}\mu_{W}\ &=&\imath^{*}\mu_{S}\ &=&\hat{\imath^{*}\mu_{S}}\left(id_{S\left(C\right)}\right).\ \end{eqnarray*}

But $\imath^{*}\mu_{S}$ is pseudo-colimiting, so $\hat{\imath^{*}\mu_{S}}$ is an isomorphism, hence $$\phi\left(C\right) \circ \psi=id_{S\left(C\right)}.$$ Therefore, $\phi$ is an isomorphism. Since pseudo-colimits are stable under isomorphisms, it follows that $j\left(S\right)$ is a pseudo-colimit for $j\circ F$.

3 added 96 characters in body

It suffices to show this for psuedo-colimits, i.e. ones for which the induced equivalence of categories is in fact an isomorphism of categories (such a strict model for the weak colimit exists, since we can explicitly calculate one in the 2-category of groupoids). Denote pseudo-colimits by $pcolim$.

Let $j$ denote the canonical $2$-functor from strict presheaves in groupoids into weak ones.

Let $F:J \to Psh\left(C,Gpd\right)$ be any pseudo-functor. Let $$\mu_{S}:F \Rightarrow \Delta_{pcolim F}$$ be a pseudo-colimiting cocone for $F$ and $$\mu_{W}:j \circ F \Rightarrow \Delta_{pcolim j \circ F}$$ be a pseudo-colimiting cocone for $j \circ F$. To simplify notation, let $S:=pcolim F$ and $W:=pcolim j\circ F$. Then $$j\mu_{S}:j\circ F \Rightarrow \Delta{j\left(S\right)}$$ is a cocone for $j \circ F$ with vertex $j\left(S\right)$. Hence there exists a morphism $\phi:W \to S$ such that $j\mu_{S}=\Delta_{\phi} \circ \mu_{W}$. We claim that $\phi$ is an isomorphism.

It suffices to show that for each $C \in C_0$, the map $\phi\left(C\right):W\left(C\right) \to S\left(C\right)$ is an isomorphism of groupods. Consider the inclusion of the object $C$ as a functor $$1\stackrel{\imath}{\rightarrow} C$$ from the terminal category. This induces two $2$-functors, and by abuse of notation, we will denote both by '$$\imath^{}:Psh\left(C,Gpd\right) \to Gpd$$' '$$\imath^{}:Gpd^{C^{op}} \to Gpd.$$' Both of these $2$-functors are pseudo left adjoints and are given by evaluation at the objet $C$, so clearly $\imath^{}j=\imath^{}$.

We want to show that '$$\phi\left(C\right)=\imath^{}\left(\phi\right)$$' is an isomorphism of groupoids. Since $\imath^{}$ is a pseudo left adjoint, it follows that '$$\imath^{*}\mu_{W}:\imath^{}\circ j \circ F=F\left(C\right) \Rightarrow \imath^{} \circ \Delta_{W}= \Delta_{W\left(C\right)}$$' is pseudo-colimiting. Now, since pseudo-colimits '$Psh\left(C,Gpd\right)$' are computed pointwise, it follows that '$\imath^{}\mu{S}$' is pseudo-colimiting. So, there exists a functor '$$\psi:S\left(C\right) \to W\left(C\right)$$' such that '$$\Delta{\psi}\circ\imath^{}\mu_{S}=\imath^{*}\mu_{W}.$$'

Notice that $$\imath^{*}\mu_{S}=\imath^{*}\left(\Delta_{\phi}\right)\circ \mu_{W})=\Delta_{\phi\left(C\right)}\circ \imath^{*}\mu_{W}.$$ So

$$\Delta_{\psi}\circ \Delta{\phi\left(C\right)}\circ \imath^{*}\mu_{W}=\imath^{*}\mu_{W}.$$ The left hand side of this equation is equal to $\hat{\imath^{*}\mu_{W}}\left(\psi \circ \phi\left(C\right)\right)$ whereas the right hand side is equal to $\hat{\imath^{*}\mu_{W}}\left(id_{W\left(C\right)}\right).$ But $\imath^{*}\mu_{W}$ is pseudo-colimiting, so $\hat{\imath^{*}\mu_{W}}$ is an isomorphism of categories, hence $\psi \circ \phi\left(C\right)=id_{W\left(C\right)}.$

Notice further that

\begin{eqnarray*} \imath^{*}\mu_{S}\left(\phi\left(C\right) \circ \psi\right)&=&\Delta_{\phi\left(C\right)} \circ \Delta{\psi} \circ \Delta{\phi\left(C\right)} \circ \imath^{*}\mu_{W}\ &=&\Delta_{\phi\left(C\right)} \circ \imath^{*}\mu_{W}\ &=&\imath^{*}\mu_{S}\ &=&\hat{\imath^{*}\mu_{S}}\left(id_{S\left(C\right)}\right).\ \end{eqnarray*}

But $\imath^{*}\mu_{S}$ is pseudo-colimiting, so $\hat{\imath^{*}\mu_{S}}$ is an isomorphism, hence $$\phi\left(C\right) \circ \psi=id_{S\left(C\right)}.$$ Therefore, $\phi$ is an isomorphism. Since pseudo-colimits are stable under isomorphisms, it follows that $j\left(S\right)$ is a pseudo-colimit for $j\circ F$.