# $F_2\circ F_1\dashv G_1\circ G_2$ and $F_2\dashv G_2$ but not $F_1\dashv G_1$

What are examples of pairs of adjoint functors the form $$F_2\circ F_1\dashv G_1\circ G_2$$ (where $$F_1$$ and $$G_1$$ are functors between, say, $$C$$ and $$D$$, and $$F_2$$ and $$G_2$$ are functors between $$D$$ and $$E$$) such that one of $$F_1\dashv G_1$$ and $$F_2\dashv G_2$$ is false (and one of these statements is true)?

I thought maybe the following adjunction from sheaf theory is an example: the inclusion functor $$\mathrm{sheaf}\to\mathrm{presheaf}$$ is right adjoint to $$\mathrm{presheaf}\xrightarrow{(-)^+}\mathrm{separatedPresheaf}\xrightarrow{(-)^+}\mathrm{sheaf},$$ where $$(-)^+$$ denotes the plus construction. Is it true that $$\mathrm{separatedPresheaf}\xrightarrow{(-)^+}\mathrm{sheaf}$$ is left adjoint to the inclusion $$\mathrm{sheaf}\to\mathrm{separatedPresheaf}$$ (while $$\mathrm{presheaf}\xrightarrow{(-)^+}\mathrm{separatedPresheaf}$$ is not left adjoint to the inclusion $$\mathrm{separatedPresheaf}\to\mathrm{presheaf}$$, as remarked on nLab)?

Your example is correct. To see this, first note that the inclusion of $$\mathbf{Sheaf}$$ into $$\mathbf{separatedPresheaf}$$ has a left adjoint given by the sheafification functor (this follows by the universal property of sheafification). It then remains to show that on separated presheaves, the functor $$(-)^+$$ agrees with the sheafification, or in other words that $$F^+ \simeq F^{++}$$ for every separated presheaf $$F$$. But this follows from $$F^+$$ being a sheaf in this case.