Terminology of locally small categories without replacement

[Colin McLarty <colin.mclarty@case.edu>]

Locally small categories are always defined as categories such that:

LS) for any objects A,B there is a set of all arrows A-->B.

When the base set theory includes the axiom scheme of replacement that
is equivalent to a prima facie stronger property:

??) for any set of objects there is a set of all arrows between them.

These two are not equivalent in the absence of the axiom scheme of
replacement.  There the second is much stronger, but it remains
important.  Is there a good term for it?

thanks, Colin


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