We say that a category $\mathbf A$ is fully embeddable into $\mathbf B$ that means that there exists a full embedding $\mathbf A\rightarrow \mathbf B$.

Then we know that each category of the form $\mathbf A\mathbf l\mathbf g(\Omega)$ is fully embeddable into each of the following constructs:

$1)$ $ \mathbf S \mathbf g\mathbf r$,

$2)$ $ \mathbf R \mathbf e\mathbf l$,

$3)$ $ \mathbf A \mathbf l \mathbf g(1,1)$, i.e., the construct of unary algebras on two operations.

But does there exist a category $\mathbf A$ such that every category (not quasicategories) can be fully embedded into $\mathbf A$?