Pullbacks of a family of arrows

[Ohad Kammar <ohad.kammar@ed.ac.uk>]

What is the standard name for the limit of the diagram consisting of a morphism
f : A --> C
and an I-indexed family of morphisms
g_i : B_i --> C
where I is a set?

This is a slight generalisation of pullbacks. I am really interested
in the terminology for the following notion:

A class of morphisms M is stable under such generalised pullbacks if,
for all such limiting cones
a : P --> A
b_i : P --> B_i
if g_i are all in M, then so is the morphism a : P --> A

(This is a slight generalisation of stability of M under pullbacks.)

Thanks,
Ohad.

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