ScalingStacks

Lemma 7.4.36. Let $M$ be a subset of $Z$ and let $Z^{\prime}$ be a subcurve of $Z$. Assume that given an admissible homotopy class of paths $\zeta$ in $Z$ with endpoints in $M$, there is an admissible path $\gamma$ in $\zeta$ contained in $Z-Z^{\prime}$. There is a faithful functor of differential pointed categories

	$\displaystyle{\mathcal{S}}^{\bullet}_{M}(Z)\wedge{\mathcal{S}}^{\bullet}(Z^{\prime})$	$\displaystyle\to{\mathcal{S}}^{\bullet}_{M\cup Z^{\prime}}(Z)$
	$\displaystyle(S,T)$	$\displaystyle\mapsto S\sqcup T$
	$\displaystyle(\alpha,\beta)$	$\displaystyle\mapsto\alpha\boxtimes\beta=(\alpha\boxtimes\operatorname{id}\nolimits)\cdot(\operatorname{id}\nolimits\boxtimes\beta)=(\operatorname{id}\nolimits\boxtimes\beta)\cdot(\alpha\boxtimes\operatorname{id}\nolimits).$