I am learning sheaf theory from Sheaves in Geometry and Logic by MacLane and Moerdijk, which I'll refer simply as [MM].
I am curious to know whether the inverse image sheaf functor has a left adjoint?
I think the answer is yes; however, I am not sure about it. Also, I couldn't find this in [MM], at least till the end of Chapter II. So given a continuous map $(X,\mathcal O_X) \xrightarrow f (Y,\mathcal O_Y)$, then my strategy is as follows:
First use the ''change of base'' result (Theorem 4 on [page 59, MM]) on the category $\mathbf{Etale} (X)$ (i.e., the category of étalé bundles on $X$) to construct a left adjoint (denoted $\Sigma_f$) to the pullback functor $\mathbf{Etale}(Y) \xrightarrow {f^*} \mathbf{Etale}(X)$. Here, we note that the category $\mathbf {Etale}(X)$ has pullbacks (to prove this we need pullbacks from the category of topological spaces and the property that pullback of an étalé bundle along a continuous map is again an étalé bundle).
Second use the functors $\mathbf {Top}/X \xrightarrow{\Gamma_X} \mathbf{Sets}^{\mathcal O_X^\text{op}}$ and $\mathbf{Sets}^{\mathcal O_X^\text{op}} \xrightarrow{\Lambda_X} \mathbf {Top}/X$ to define $\mathbf{Sh}(X) \xrightarrow{L_f} \mathbf{Sh}(Y)$ as $L_f=\Gamma_Y\Sigma_f\Lambda_X$.
So it remains to show that $L_f \dashv f^*$, where $\mathbf{Sh}(Y) \xrightarrow{f^*} \mathbf{Sh}(X)$ is given by the composition $\Gamma_Xf^*\Lambda_Y$. For this we derive,
\begin{align} \hom (\Gamma_Y \Sigma_f \Lambda_X F, G) \cong&\ \hom (\Gamma_Y \Sigma_f \Lambda_X F, \Gamma_Y \Lambda_Y G)\\ \cong&\ \hom (\Lambda_Y \Gamma_Y \Sigma_f \Lambda_X F, \Lambda_Y G)\\ \cong&\ \hom (\Sigma_f \Lambda_X F, \Lambda_Y G)\\ \cong&\ \hom (\Lambda_X F, f^* \Lambda_Y G)\\ \cong&\ \hom (F, \Gamma_X f^* \Lambda_Y G). \end{align}
Are the above steps valid? If yes, is there a special name to the functor $L_f$?