Lemma 7.28.6. Assume given sites $\mathcal{C}', \mathcal{C}, \mathcal{D}', \mathcal{D}$ and functors

\[ \xymatrix{ \mathcal{C}' \ar[r]_{v'} \ar[d]_{u'} & \mathcal{C} \ar[d]^ u \\ \mathcal{D}' \ar[r]^ v & \mathcal{D} } \]

Assume

$u$, $u'$, $v$, and $v'$ are cocontinuous giving rise to morphisms of topoi $f$, $f'$, $g$, and $g'$ by Lemma 7.21.1,

$v \circ u' = u \circ v'$,

$v$ and $v'$ are continuous as well as cocontinuous, and

for any object $V'$ of $\mathcal{D}'$ the functor ${}^{u'}_{V'}\mathcal{I} \to {}^{\ \ \ u}_{v(V')}\mathcal{I}$ given by $v$ is cofinal.

Then $f'_* \circ (g')^{-1} = g^{-1} \circ f_*$ and $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$.

**Proof.**
The categories ${}^{u'}_{V'}\mathcal{I}$ and ${}^{\ \ \ u}_{v(V')}\mathcal{I}$ are defined in Section 7.19. The functor in condition (4) sends the object $\psi : u'(U') \to V'$ of ${}^{u'}_{V'}\mathcal{I}$ to the object $v(\psi ) : uv'(U') = vu'(U') \to v(V')$ of ${}^{\ \ \ u}_{v(V')}\mathcal{I}$. Recall that $g^{-1}$ is given by $v^ p$ (Lemma 7.21.5) and $f_*$ is given by ${}_ su = {}_ pu$. Hence the value of $g^{-1}f_*\mathcal{F}$ on $V'$ is the value of ${}_ pu\mathcal{F}$ on $v(V')$ which is the limit

\[ \mathop{\mathrm{lim}}\nolimits _{u(U) \to v(V') \in \mathop{\mathrm{Ob}}\nolimits ({}^{\ \ \ u}_{v(V')}\mathcal{I}^{opp})} \mathcal{F}(U) \]

By the same reasoning, the value of $f'_*(g')^{-1}\mathcal{F}$ on $V'$ is given by the limit

\[ \mathop{\mathrm{lim}}\nolimits _{u'(U') \to V' \in \mathop{\mathrm{Ob}}\nolimits ({}^{u'}_{V'}\mathcal{I}^{opp})} \mathcal{F}(v'(U')) \]

Thus assumption (4) and Categories, Lemma 4.17.4 show that these agree and the first equality of the lemma is proved. The second equality follows from the first by uniqueness of adjoints.
$\square$

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