What are the "objects of a language"? The entities its first-order quantifiers run over? [That's the usual story!]
Category theory is usually set up with a two-sorted first order language with objects and morphisms as distinct sorts. But it would only be mildly perverse to take the official language of category theory to be single-sorted, with quantifiers running over entities comprising the category's objects and the morphisms (then restricting quantifiers when you want to talk about just the objects, or just the morphisms).
In this single-sorted version, the objects in the single domain of the language will include morphisms between [some of] the domain's own objects.
Presumably, though, you are not interested in this triviality! So I guess the question needs to be refined to give us something clearer to grapple with (so we know whether slice or arrow categories are relevant to real your concerns, for example.)
The real question here is what properties the language of category theory captures.
A statement in the (finitary) language of category theory is one formed from propositions of the form $f = g$ where $f, g : X \to Y$ using the propositional operations of $\lor, \land, \top, \bot, \neg, \implies$, together with quantifiers either over all objects (eg $\forall X$, $\exists X$ where $X$ is an object variable) or over all morphisms between two given objects (eg $\forall f : X \to Y$ or $\exists f : X \to Y$ where $X, Y$ are objects).
Two things can be shown about statements in the language of category theory. Suppose given a statement $\phi(X_1, ..., X_n, f_1, ..., f_m)$ in the language of category theory with object variables $X_1, ..., X_n$ and free function variables $f_i : X_{d_i} \to X_{c_i}$ for $1 \leq i \leq m$.
Now consider a category and two different variable assignments - one assignment $X_1 \mapsto W_1, ..., X_n \mapsto W_n$ where each $W_i$ is an object, and $f_1 \mapsto g_1, ..., f_m \mapsto g_m$ where $g_i : W_{d_i} \to W_{c_i}$ for all $i$, and another assignment $X_1 \mapsto Y_1, ..., X_n \mapsto Y_n$ and $f_1 \mapsto h_1, ..., f_m \mapsto h_m$, where $Y_i$ is an object for all $i$ and $h_j : W_{d_j} \to W_{c_j}$ for all $j$. Suppose there are isomorphisms $k_1 : W_1 \to Y_1, ..., k_n : W_n \to Y_n$ such that for all $i$, $h_i = k_{c_i} \circ f_i \circ k_{d_i}^{-1}$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m) \iff \phi(Y_1, ..., Y_n, h_1, ..., h_m)$. This is known as "isomorphism invariance of truth".
Now consider a functor $F : C \to D$ which is fully faithful and essentially surjective, and a variable assignment in $C$ of the form $X_i \mapsto W_i$, $f_j \mapsto g_j$ as above. We assume that $\phi$ has no free variables other than the $X_i$ and $f_i$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m)$ iff $\phi(F(W_1), ..., F(W_n), F(g_1), ..., F(g_m))$. This is known as "equivalence invariance of truth".
Both of the above can be proved by induction on formulas.
The two above statements can be generalised to the infinitary language of category theory, which allows quantification over external sets, though some care must be taken in settings without the axiom of choice.
So the language of category can only discuss properties which are invariant up to isomorphism and up to equivalence of categories. No finer-grained discrimination is possible.
For example, consider the category of sets. It follows from isomorphism invariance that the language of category cannot distinguish two sets which have the same cardinality. The language of category cannot distinguish a monic $f : S \to T$ from a subset $S \subseteq T$.
For the category of topological spaces with morphisms the continuous maps quotiented by homotopy, the language of category theory cannot distinguish between homotopy equivalent spaces.
So the key when using category theory is to find the right amount of data for morphisms to carry based on the underlying subject one is studying and the particular context.
Best Answer
Notice that, from the perspective of category theory, the property you propose is not very natural, because it is not invariant under equivalence. However, we may rephrase it to be more natural by instead asking for an object $\zeta$ such that every morphism $\zeta \to X$ is an isomorphism. These are known as maximal objects. One common class of examples is given by strict terminal objects.