Unit isomorphism in SVECT

Question

What is the unit isomorphism $$X\otimes\mathbb{C}^{1|0}\cong X$$ in the monoidal caregory of super-vector spaces? Is it $$x\otimes\lambda\mapsto \lambda x$$ like in the monoidal category of vector spaces?

Answer 1

Since a super-vector space can be regarded as a vector space graded over $\mathbb{Z}_2$, I'd better say that the isomorphism $$ (X_0\oplus X_1)\otimes(\mathbb C\oplus 0) \cong X_0\oplus X_1 $$ results from the sum of the isomorphisms $\eta_0:X_0\otimes\mathbb C \cong X_0$ and $\eta_1:X_1\otimes\mathbb C\cong X_1$: $$ (X_0\oplus X_1)\otimes(\mathbb C\oplus 0) = (X_0\otimes \mathbb C)\oplus (X_1\oplus \mathbb C)\xrightarrow{\eta_0\oplus \eta_1} X_0\oplus X_1 $$ (I'm using implicitly that the super-$K$-linear tensor product $V\otimes_\text{s} W$ is such that $(V\otimes_\text{s} W)_\epsilon = \bigoplus_{i+j=\epsilon\pmod{2}} V_i\otimes_K W_j$)