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Merge branch 'master' of github.com:UniMath/SymmetryBook
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marcbezem committed Jan 16, 2025
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Expand Up @@ -206,17 +206,20 @@ \section{The pullback}

\begin{xca}
Prove that if $f:\Hom(H,G)$ and $f':\Hom(H',G)$ are homomorphisms,
then the pointed version of \cref{xca:univpropofpullback} induces an equivalence
$$
\Hom(K,H)\times_{\Hom(K,G)}\Hom(K,H')\simeq \Hom(K,H\times_GH')
$$
for all groups $K$ and an equivalence
then the pointed version of \cref{xca:univpropofpullback} induces an
equivalence
\[
\Hom(K,H)\times_{\Hom(K,G)}\Hom(K,H')\simeq \Hom(K,H\times_GH')
\]
for all groups $K$ and an equivalence%
\stepcounter{footnote}\footnotetext{%
Hint: set $A\defequi \Sc$, $B\defequi \BH$, $C\defequi \BH'$ and $D\defequi \BG$.}%
\addtocounter{footnote}{-1}
\[
\USymH \times_{\USymG} \USymH'
\simeq (\shape_{H\times_GH'}=\shape_{H\times_GH'}).\text{\footnotemark}
\]
Elevate the last equivalence to a statement about abstract groups.\footnotetext{%
Hint: set $A\defequi \Sc$, $B\defequi \BH$, $C\defequi \BH'$ and $D\defequi \BG$.}
Elevate the last equivalence to a statement about abstract groups.
\end{xca}

\begin{remark}
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