categories: Re:Time for functors to grow up; three queries

[Steve Lack <stevel@maths.usyd.edu.au>]

In my earlier posting I sometime seem to have left out
the most important bit:

 > A functor F: A --> C is said to
 > factor through its \emph{essential image}, E, as
 > 
 >                  e      i
 >               A ---> E ---> C
 > 
 > when e is eso and full, i is injective on objects and also faithful,
 > and under the action of i, E is a replete subcategory of C.
 > 
 > QUERY (1):  Is (4) stated properly?  Is there any reference into the
 >             literature at all for this definition?  I looked fairly
 >             intensively this past weekend and could locate nothing...
 > 
 > 
 > Propositions.
 > (1) Each functor in the category of categories has an essential image factorization.
 > (2) The composition of two eso functors is again eso.
 > (3) For each eso functor F: A ---> C
 >     with image factorization through its essential image
 > 
 >                  e      i
 >               A ---> E ---> C
 > 
 >     the functor  i  makes E an essential subcategory of C
 >     and indeed,  i  is a bijection on objects.
 >     Furthermore  e  is both full and faithful.
 > 

Such factorizations need not exist. Let A be the category with 
four objects a,b,b',c and two non-identity arrows f:a-->b and g:b'-->c;
let C be the category with three objects a,b,c, and non-identity 
arrows f:a-->b, g:b-->c, and gf:a-->c. The evident functor F:A-->C
sending f to f and g to g has no such factorization, since it factorizes
through no proper subcategory of C but is not itself full.

Steve.