The subgroup inclusions $$ U(N) \hookrightarrow Sp(N):=U(2N) \cap Sp(2N,\mathbb{C}), \quad U(N) \hookrightarrow O(2N) $$
induces some morphisms $$ f_1: U(\infty)\to Sp(\infty), f_2: U(\infty)\to O(\infty). $$.
Are they related by the Bott periodicity isomorphisms $U(\infty) \simeq \Omega^{4}U(\infty)$ and $Sp(\infty) \simeq \Omega^{4} O(\infty)$? That is, is the map
$$ U(\infty) \xrightarrow{\sim} \Omega^{4}U(\infty) \xrightarrow{\Omega^{4}f_{2}} \Omega^{4}O(\infty) \xrightarrow{\sim} Sp(\infty) $$
homotopic to $f_{1}$?
The motivation for my question is to compute the map $\pi_5[U(\infty)] = \mathbb{Z} \to \mathbb{Z}_{2} = \pi_5[Sp(\infty)]$ induced by $f_1$. If the above is true, then using
$$ [S^5, Sp(\infty)] \simeq [\Sigma^4 S^1, Sp(\infty)] \simeq [S^1, \Omega^4 Sp(\infty)] \simeq [S^1, O(\infty)], $$
one would be able to reduce the problem to computing $\pi_1[U(\infty)] \to \pi_1[O(\infty)]$ induced by $f_2$, which seems to me just as $\mod 2: \mathbb{Z} \to \mathbb{Z}_{2}$.