Let $$f\colon X\wedge Y\to Z$$ a morphism of (symmetric) spectra with adjoint morphism $$g\colon X\to [Y,Z]$$ where the brackets denote the internal hom of spectra. Let $\tau_{\leq n}$ be the standard truncation functor (obtained by glueing in all the maps from $>n$-spheres such that the truncated spectrum $\tau_{\leq n}X$ has no stable homotopy groups above $n$ and the same stable homotopy groups as $X$ in degrees $\leq n$.). Fix an $n$.

I apologize if my question is dumb.

Suppose $$\tau_{\leq n} (g)\colon \tau_{\leq n}X\to \tau_{\leq n}[Y,Z]$$ is homotopic to the zero map, does is follow that $$ \tau_{\leq n}(f)\colon \tau_{\leq n}(X\wedge Y)\to \tau_{\leq n}Z $$ is homotopic to the zero map?

Is there any literature on the relation of the truncation to the smash-internal hom adjunction or isn't there anything reasonable to expect?


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