Jean asks: Is there a standard name for the squares where the canonical map is monic , i.e. the pair of maps A --->B and A --->C is jointly monic. In the early 60s at the annual AMS meeting held at Denver, Eilenberg, Mac Lane and I sat down to "settle" the terminology. ("Denver One" I called it.) There were just two things we totally agreed on: "weak" is the operator on definitions that removes uniqueness conditions and "partial" the operator that removes existence conditions. So the answer to Jean's question would be "partial pullback". As for the other side -- when the pair of maps are jointly epic -- I've seen them called "near-pullbacks" in the theoretical computer science community. Functors between regular categories that preserve near-pullbacks are precisely those that preserve "weak tabulations" of (n-ary) relations.