ct.category theory - Monoidal operations on categories where the maps on Aut, End are injective

Your conditions don't seem to obtain very often, unfortunately.

Let's begin with the one-object case (where the one object is the unit, as it must be). This is the same thing as a monoid object in monoids, or equivalently, a commutative monoid $A$, by Eckmann-Hilton. Your requirement (1) appears to be that the multiplication $Atimes Ato A$ is injective; requirement (2) is the same, applied to the invertible elements. This says that every element of $A$ can be uniquely factored, so that $A$ is trivial. So in the one-object case, (1) happens exactly once, and (2) is equivalent to the nonexistence of nontrivial invertible elements.

Using this, we deduce that the unit in a monoidal category satisfying either (1) or (2) is either very rigid in the sense that it admits no nontrivial endomorphisms or, respectively, rigid in the sense that it admits no nontrivial automorphisms.

If the category $mathcal{C}$ has an initial object $varnothing$ that is preserved by the monoidal structure (so that $Xotimesvarnothing=varnothingotimes X=varnothing$ for all $Xinmathcal{C}$), then every object has to (1) very rigid or (2) rigid. This kills off the bulk of the "algebraic" examples (like modules over a ring, etc.)

Also, note that your conditions are symmetric, i.e., (1) or (2) holds for $mathcal{C}$ iff it holds for $mathcal{C}^{mathrm{op}}$. So Reid's example showing that not all cartesian categories satisfy (2) also works to show that not all cocartesian categories satisfy (2). (We contemplate the coproduct in $mathrm{Set}^{mathrm{op}}$...)

I don't want to be overly negative; so here's a situation in which I think your conditions do hold: suppose $mathcal{C}$ a category with all finite coproducts. Suppose the morphism $Xto Xsqcup Y$ is monic for any $X,Yinmathcal{C}$. Then $mathcal{C}$ (equipped with the coproduct) satisfies your condition. (So this works for finite sets with disjoint union, for instance.) Dually, if $mathcal{C}$ is a category with all finite products such that the morphism $Xtimes Yto X$ is epic for any $X,Yinmathcal{C}$, then $mathcal{C}$ (equipped with the product) satisfies your condition. (So this works for the product on nonempty finite sets, for instance.)