Definition 0.1   A category $\mathcal{C}_1$ with coproducts is called a $C_1$ -category if for every family of of monomorphisms $\left\{u_i: A_i \to B_i\right\}$ the morphism $$\iota := \oplus_i \, u_i: \oplus_i \, A_i \to \oplus_i \, B_i $$ is also a monomorphism ([1]).

Remark 0.1   With certain additional conditions (as explained in ref. [1]) $\mathcal{C}_1$ may satisfy the Grothendieck axiom $\mathcal{A}b5$ , thus becoming a $C_3$ -category (Ch. 11 in [1]).