This is a crosspost of this MSE question.

The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$) form a well-known adjunction for $\mathsf{Set}$-valued sheaves (I think also for sheaves valued in algebraic categories).

Is there anything interesting to be said about $f^\ast,g^\ast$ and $f_\ast,g_\ast$ when $f\simeq g$? What about $f_!,g_!$ and $f^!,g^!$?