Left adjoint to inverse image of sheaves We write $\mathrm{Sh}(X)=\mathrm{Sh}(X, \mathsf{Sets})$ for the category of sheaves of sets on $X$. 
In this situation, if $f: X \to Y$ is an etale map (i.e. a local homeomorphism) of topological spaces then we can show that there is a left adjoint $f_!: \mathrm{Sh}(X) \to \mathrm{Sh}(Y)$ to the inverse image functor $f^{-1}: \mathrm{Sh}(Y) \to \mathrm{Sh}(X)$, which corresponds to the dependent sum $\Sigma_f$ of etale spaces of sheaves.
I am curious if this left adjoint exists if $\mathsf{Sets}$ is replaced by some other complete and cocomplete category $\mathsf{C}$? 
If $j: U \hookrightarrow Y$ is the inclusion of an open subset, then $j_!$ exists, and is called the extension by $0$, where $0$ is the initial object in $\mathsf{C}$.
Does a left adjoint $f_!$ still exist if we replace $j: U \hookrightarrow X$ by an arbitary etale map $f: X \to Y$?
 A: Yes, the left adjoint do exist if the category is cocomplete and has the solution set criterion, for example if $\mathbf{C}$ is Grothendieck abelian. This follows from the adjoint functor theorem.
Indeed, $f^{-1}$ commutes with finite projective limits, so we just need to check that $f^{-1}$ commute with infinite products. But this can be checked locally since $j^{-1}$ do have a left adjoint for $j:U\rightarrow X$ an open immersion.
More precisely, let $\prod \mathcal{F}_i$ be an arbitrary product of sheaves on $Y$. We need to prove that the natural map $f^{-1}\prod\mathcal{F}_i\rightarrow\prod f^{-1}\mathcal{F}_i$ is an isomorphism. For this, it is enough to prove that this is an isomorphism when restricted on some open cover. So write $X=\bigcup U_j$ where $f_j:U_j\rightarrow X\rightarrow Y$ is an open immersion. Then $(f^{-1}\prod\mathcal{F}_i)_{|U_j}= f_j^{-1}\prod \mathcal{F}_i=\prod f_j^{-1}\mathcal{F}_i=(\prod f^{-1}\mathcal{F}_i)_{|U_j}$.

In fact, we have the following construction for $f_!\mathcal{F}$ at least in the case of sheaves of set, abelian groups and $\mathcal{O}_X$-modules. This is the sheafification of the presheaf
$$V\subset Y\mapsto \coprod_{s\in Map_Y(V,X)}\mathcal{F}(s(V))$$
where $Map_Y(V,X)$ denote the set of sections $V\rightarrow X$ (a section will necessarily be an homeomorphism onto an open subset of $X$), and the coproduct is taken in the corresponding category (sets, abelian group, $\mathcal{O}_X(V)$-modules).
(Maybe this formula works in arbitrary cocomplete categories, I am not sure...)

Finally, there is another reason that this works : the category of sheaves on $X$ is equivalent to the category of sheaves on $Et/X$. Then this is the usual extension by zero functor on sheaves on site. 
