A tensor-free comparison used in the finite certificate

The next theorem is not a construction of a net. It compares state restrictions through already declared inclusions.

Let $M_{\mathrm{in}},M_{\mathrm{out}}$ be unital $C^{\ast }$-algebras. Let

be unital injective $\ast$-homomorphisms, and $T:M_{\mathrm{out}}\to M_{\mathrm{in}}$ be UCP.