Recall that an operator $T:X\rightarrow Y$ between Banach spaces is called strictly singular if if it is not an isomorphism when restricted to any infinite-dimensional closed subspace of $X$. An operator $T:X\rightarrow Y$ is said to be strictly cosingular provided that for no infinite-dimensional Banach space $Z$ there exist surjective operators $R:X\rightarrow Z$ and $S:Y\rightarrow Z$ such that $R=ST$. It is elementary that an operator $T$ is strictly cosingular (strictly singular) whenever its adjoint $T^{*}$ is strictly singular (strictly cosingular, respectively). My question is the converse to this basic fact. More precisely, for an operator $T:X\rightarrow Y$,

Question 1. Under what conditions on $X$, the adjoint $T^{*}$ is strictly cosingular whenever $T$ is strictly singular?

Question 2. Under what conditions on $Y$, the adjoint $T^{*}$ is strictly singular whenever $T$ is strictly cosingular?