Identity in Subcategory C': Does it Coincide with Identity in Category C?

[SVD]

Using the definition given below, I wonder whether we can deduce that for each object A in C',
the identity for A in C' coincides with the identity for A in C.

Let C' and C be two categories which satisfies that
(i)each objects in C' belongs to C
(ii)each hom-set in C' is contained in the corresponding hom-set in C.
(iii)each composition in C' is the restriction of that in C .

 

[micromass]

No. Let C and C' consist out of exactly one object A. Let

$$Hom_{C^\prime}(A,A)=\{a\}~\text{and}~Hom_C(A,A)=\{a,b\}$$

with

$$a\circ a = a\circ b=b\circ a=a~\text{and}~b\circ b=b$$