# Does a fully faithful functor apply to identity arrows?

In Saunders Mac Lane's Categories for the working mathematician one can read, when talking about fully faithful functors:

[...], but this need not mean that the functor itself is an isomorphism of categories, for there may be objects of B not in the image of T

Given a fully faithful functor $T: C \to B$ and an object together with its identity arrow $c \in C, 1_c: c \to c$, and given $T 1_c = 1_{Tc}: T c \to T c$ in $B$, how can there be a fully faithful functor between categories with unequal amounts of elements?

Because faithful functors follow the rule $Tf_1 = Tf_2 \Rightarrow f_1 = f_2$ and with categories with unequal (or rather less) elements there must be some $1_{Tc}$ that is equal to some $1_{Tc'}$

For me it seems to be the same reason why there is no injective function $\mathbb{Z} \to \mathbb{N}$

• For any functor $T1_c=1_{Tc}$. Oct 1 '17 at 18:06
• Functors map identity morphisms to identity morphisms by definition. Oct 1 '17 at 18:08
• Further to the above comments, here's a silly class of examples of full and faithful functors between categories with unequal numbers of objects. Let $\mathcal{C}$ be any indiscrete category, i.e. a set of objects such that between any two objects there is exactly one morphism, and let $\mathbf{1}$ be the terminal category, i.e. the category with one object and one (identity) morphism. The unique functor $\mathcal{C} \to \mathbf{1}$ is full and faithful. Oct 1 '17 at 18:12
• @CliveNewstead Really? A faithful functor implies $Tf_1 = Tf_2 \Rightarrow f_1 = f_2$, but if $\forall f \in \mathcal{C}. T f = id_1$, (where $id_1$ is the single arrow in the terminal category) then this doesn't hold. Oct 1 '17 at 20:24
• Because $Tf_1= Tf_2 \implies f_1=f_2$ need not hold, even if $T$ is faithful. For it to hold, you have to assume first that $f_1, f_2$ have the same domain and codomain: a faithful functor is injective when restricted to $Hom(A,B)$, not when restricted to $Mor(C)$ Oct 1 '17 at 20:55

Let $\mathcal C$ be any full subcategory of a category $\mathcal D$. Then the inclusion $\mathcal C \subseteq \mathcal D$ gives a functor $F\colon\mathcal C \to \mathcal D$. This functor is always fully faithfull, but it is not always true that a subcategory is isomorphic to the larger category.