Does the following exactness property have a name?

Consider a category that has pullbacks, and colimits of countable sequences of monomorphisms. Suppose given a diagram

such that each $A_n \to A_{n+1}$ is monic, the bottom row is a colimit, and all the squares

are pullbacks (hence each $B_n \to B_{n+1}$ is also monic). Then the exactness property says that the top row is a colimit if and only if all the squares

are pullbacks.