Let $Gpd$ denote the category of groupoids and functors. Let $Gpd\_{con}$ denote the subcategory spanned by connected groupoids, i.e for every $x,y\in Ob(Gpd\_{con})$, there is at least one morphism $x\rightarrow y$.


Does the canonical inclusion $i:Gpd\_{con}\rightarrow Gpd$ have a left adjoint? If so, does it preserve products?



Left adjoint to inclusion of Connected Groupoids into Groupoids
No, it does not. If it did, then $\mathrm{Gpd}\_{\mathrm{con}}$, like any reflective subcategory, would be closed under limits in $\mathrm{Gpd}$. But it is not closed under equalizers. For instance, let $X$ be the contractible groupoid with two objects $x,y$, and $G$ any nontrivial group regarded as a connected groupoid, let $f:X\to G$ send the isomorphism $x\cong y$ to the identity of $G$, and let $g:X\to G$ send it to some nonidentity element of $G$. Then the equalizer of $f$ and $g$ in $\mathrm{Gpd}$ is the discrete groupoid on two objects, which is no longer connected.


