# The Stacks Project

## Tag 0CLF

Lemma 86.34.5. Assume given a $2$-commutative diagram $$\xymatrix{ \mathop{\rm Spec}(K) \ar[r]_-x \ar[dd]_j & \mathcal{X} \ar[d]^f \\ & \mathcal{Y} \ar[d]^g \\ \mathop{\rm Spec}(A) \ar[r]^-z & \mathcal{Z} }$$ Choose a $2$-arrow $\gamma : z \circ j \to g \circ f \circ x$. Let $\mathcal{C}$ be the category of dotted arrows for the outer rectangle and $\gamma$. Let $\mathcal{C}'$ be the category of dotted arrows for the square $$\xymatrix{ \mathop{\rm Spec}(K) \ar[r]_-{f \circ x} \ar[d]_j & \mathcal{Y} \ar[d]^g \\ \mathop{\rm Spec}(A) \ar[r]^-z & \mathcal{Z} }$$ and $\gamma$. There is a canonical functor $\mathcal{C} \to \mathcal{C}'$ which turns $\mathcal{C}$ into a category fibred in groupoids over $\mathcal{C}'$ and whose fibre categories are categories of dotted arrows for certain squares of the form $$\xymatrix{ \mathop{\rm Spec}(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\ \mathop{\rm Spec}(A) \ar[r]^-y & \mathcal{Y} }$$ and some choice of $y \circ j \to f \circ x$.

Proof. Omitted. Hint: If $(a, \alpha, \beta)$ is an object of $\mathcal{C}$, then $(f \circ a, \text{id}_f \star \alpha, \beta)$ is an object of $\mathcal{C}'$. Conversely, if $(y, \delta, \epsilon)$ is an object of $\mathcal{C}'$ and $(a, \alpha, \beta)$ is an object of the category of dotted arrows of the last displayed diagram with $y \circ j \to f \circ x$ given by $\delta$, then $(a, \alpha, (\text{id}_g \star \beta) \circ \epsilon)$ is an object of $\mathcal{C}$. $\square$

The code snippet corresponding to this tag is a part of the file stacks-morphisms.tex and is located in lines 7755–7786 (see updates for more information).

\begin{lemma}
\label{lemma-cat-dotted-arrows-composition}
Assume given a $2$-commutative diagram
$$\xymatrix{ \Spec(K) \ar[r]_-x \ar[dd]_j & \mathcal{X} \ar[d]^f \\ & \mathcal{Y} \ar[d]^g \\ \Spec(A) \ar[r]^-z & \mathcal{Z} }$$
Choose a $2$-arrow $\gamma : z \circ j \to g \circ f \circ x$.
Let $\mathcal{C}$ be the category of dotted arrows for
the outer rectangle and $\gamma$. Let $\mathcal{C}'$ be the
category of dotted arrows for the square
$$\xymatrix{ \Spec(K) \ar[r]_-{f \circ x} \ar[d]_j & \mathcal{Y} \ar[d]^g \\ \Spec(A) \ar[r]^-z & \mathcal{Z} }$$
and $\gamma$. There is a canonical functor $\mathcal{C} \to \mathcal{C}'$
which turns $\mathcal{C}$ into a category fibred in groupoids over
$\mathcal{C}'$ and whose fibre categories are categories of dotted arrows
for certain squares of the form
$$\xymatrix{ \Spec(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\ \Spec(A) \ar[r]^-y & \mathcal{Y} }$$
and some choice of $y \circ j \to f \circ x$.
\end{lemma}

\begin{proof}
Omitted. Hint: If $(a, \alpha, \beta)$ is an object of $\mathcal{C}$,
then $(f \circ a, \text{id}_f \star \alpha, \beta)$ is an object
of $\mathcal{C}'$. Conversely, if $(y, \delta, \epsilon)$ is an
object of $\mathcal{C}'$ and $(a, \alpha, \beta)$ is an object
of the category of dotted arrows of the last displayed diagram
with $y \circ j \to f \circ x$ given by $\delta$, then
$(a, \alpha, (\text{id}_g \star \beta) \circ \epsilon)$ is an
object of $\mathcal{C}$.
\end{proof}

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