I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory. 


Let $\mathbb{E}=(E\_n)\_{n\in \mathbb{N}}$ be an $\Omega$-spectrum which is also a ring spectrum. That is to say, there is a map $\mu\colon \mathbb{E} \wedge \mathbb{E}\to \mathbb{E}$ satisfying the usual properties so that the cohomology defined by $\mathbb{E}$ has a cup-product 
$$
\mathbb{E}^{p}(\underline{\phantom{a}})\otimes \mathbb{E}^q (\underline{\phantom{a}})\to \mathbb{E} ^{p+q}(\underline{\phantom{a}}) \ 
 $$ 
A fact from classic homotopy theory is that $E\_0=\Omega^\infty \mathbb{E}$ is also a ring object (since the functor $\mathbb{E}^{0}(\underline{\phantom{a}})$ maps spaces to rings). Hence, there is a map $u\colon E\_0\times E\_0\to E\_0$ with usual properties representing the cup-product $\mathbb{E}^{0}(\underline{\phantom{a}})\otimes \mathbb{E}^0 (\underline{\phantom{a}})\to \mathbb{E} ^{0}(\underline{\phantom{a}})$. Since $E\_n=\Omega^\infty \mathbb{E}[n]$ represents the groups $\mathbb{E}^n(\underline{\phantom{a}})$, my question is the following:



> 
> **Question**:$$\mbox{Are there maps } u\_{p,q}\colon E\_p\times E\_q\to E\_{p+q} \,, \, p,q\in \mathbb{N}\, ,\, \mbox{ inducing the cup product}? 
> $$ $$\mbox{Maybe maps from } E\_p\wedge E\_q \ \mbox{ to } \ E\_{p+q}?$$
> 
> 
> 



Infinite loop space of ring spectra: the cup product
As Neil Strickland points out at the comments section: the existence of $u\_{p,q}$ follows from Yoneda lemma, as in the case of $u\_{0,0}$, since $E\_p\times E\_q$ represents $\mathbb{E}^p(\underline{\phantom{a}})\otimes \mathbb{E}^q(\underline{\phantom{a}})$.


