ScalingStacks

Consider $f:Z\to Z^{\prime}$ a morphism of curves and assume $f\circ\xi_{1}^{+}$ is outgoing for $Z^{\prime}$ and $f\circ\xi_{2}^{-}$ is incoming for $Z^{\prime}$.

The morphism $f$ extends uniquely to a morphism of curves $f:Z_{\xi}\to Z^{\prime}_{f\circ\xi}$.

The functor $f:{\mathcal{S}}^{\bullet}_{f,M}(Z)\to{\mathcal{S}}^{\bullet}_{f(M)}(Z^{\prime})$ can be equipped with a structure of morphism of $2$-representations $L_{\xi_{1}^{+}}^{\bullet}\to L_{f\circ\xi_{1}^{+}}^{\bullet}$ and of morphism of $2$-representations $R_{\xi_{2}^{-}}^{\bullet}\to R_{f\circ\xi_{2}^{-}}^{\bullet}$ (Lemma 8.1.7 and §8.1.5), and it induces a differential pointed functor (cf §4.3.4)

where $\lambda^{\prime}$ is the analog of the map $\lambda$ for $Z^{\prime}$.

We obtain a commutative diagram of differential pointed functors

Assume $f_{|Z}$ is strict. It follows that $f$ is strict. The functor $f^{\#}:\operatorname{add}\nolimits({\mathcal{S}}_{f(M)}(Z^{\prime}))\to\operatorname{add}\nolimits({\mathcal{S}}_{M}(Z))$ can be equipped with a structure of morphism of $2$-representations $L_{f\circ\xi_{1}^{+}}\to L_{\xi_{1}^{+}}$ and of morphism of $2$-representations $R_{f\circ\xi_{2}^{-}}\to R_{\xi_{2}^{-}}$ (Lemma 8.1.4 and §8.1.5), and it induces a differential functor (cf §4.3.4)

We obtain a commutative diagram of differential functors commuting with coproducts