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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)?

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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.

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