Lemma 13.14.3. Assumptions and notation as in Situation 13.14.1. Let $f : X \to Y$ be a morphism of $\mathcal{D}$.

If $RF$ is defined at $X$ and $Y$ then there exists a unique morphism $RF(f) : RF(X) \to RF(Y)$ between the values such that for any commutative diagram

\[ \xymatrix{ X \ar[d]_ f \ar[r]_ s & X' \ar[d]^{f'} \\ Y \ar[r]^{s'} & Y' } \]with $s, s' \in S$ the diagram

\[ \xymatrix{ F(X) \ar[d] \ar[r] & F(X') \ar[d] \ar[r] & RF(X) \ar[d] \\ F(Y) \ar[r] & F(Y') \ar[r] & RF(Y) } \]commutes.

If $LF$ is defined at $X$ and $Y$ then there exists a unique morphism $LF(f) : LF(X) \to LF(Y)$ between the values such that for any commutative diagram

\[ \xymatrix{ X' \ar[d]_{f'} \ar[r]_ s & X \ar[d]^ f \\ Y' \ar[r]^{s'} & Y } \]with $s, s'$ in $S$ the diagram

\[ \xymatrix{ LF(X) \ar[d] \ar[r] & F(X') \ar[d] \ar[r] & F(X) \ar[d] \\ LF(Y) \ar[r] & F(Y') \ar[r] & F(Y) } \]commutes.

## Comments (2)

Comment #8375 by ElĂas Guisado on

Comment #8981 by Stacks project on

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