Let me first write down the question, and then give all the relevant definitions.

> **Q.** Let $H$ and $K$ be monoids, and assume $H$ is essentially equimorphic to $K$. Is it true that $H$ is atomic if and only if so is $K$?

I can prove the "if" part of the statement, and I am buffled by the other. In any case (and an "external point of view" can be quite useful in these cases, which is why I'm asking here).

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We denote by $\mathscr{F}^\ast(\mathscr{U})$, for a fixed set $\mathscr{U}$, the free monoid with basis $\mathscr{U}$ (as usual, we assume $\mathscr{F}^\ast(\mathscr{U})$ contains $\mathscr{U}$ as a set). We use the symbol $\ast$ for the operation of $\mathscr{F}^\ast(\mathscr{U})$. Given $z \in \mathscr{U}$ and $z_1, \ldots, z_n \in \mathscr{U}$, we take
$$
\mathsf{v}_z(z_1 * \cdots * z_n) := \bigl|\bigl\{i \in [\![ 1, n ]\!]: z_i = z\bigr\}\bigr|.
$$
Then, for every $\mathfrak z \in \mathscr{F}^\ast(\mathscr{U})$ we define the *length* of $\mathfrak z$ by
$$
\|\mathfrak z\|_{\mathscr{U}} :=
\left\{
\begin{array}{ll}
0 & \text{if }\mathfrak z = 1_{\mathscr{F}^\ast(\mathscr{U})} \\
\sum_{z \in \mathscr{U}} \mathsf{v}_z(\mathfrak z) & \text{otherwise}
\end{array}
\right. .
$$
With this in hand, let $H$ be a multiplicatively written monoid. We denote by $H^\times$ the *set of units* (or (*invertible elements*) of $H$, by $\mathcal A(H)$ the *set of atoms* (or *irreducible elements*) of $H$, by $\pi_H$ the unique homomorphism $\mathscr{F}^\ast(H) \to H$ such that $\pi_H(x) = x$ for every $x \in H$,
and by $\mathscr{C}_H$ the smallest (monoid) congruence on $\mathscr{F}^\ast(\mathcal A(H))$ determined by the following condition:

 - If $\mathfrak a = a_1 \ast \cdots \ast a_m$ and $\mathfrak b = b_1 \ast \cdots \ast b_n$ are non-empty $\mathcal A(H)$-words of length $n$, then $(\mathfrak a, \mathfrak b) \in \mathscr{C}_H$ if and only if $\pi_H(\mathfrak a) = \pi_H(\mathfrak b)$, $m = n$, and $a_1 \simeq_H b_{\sigma(1)}, \ldots, a_n \simeq_H b_{\sigma(n)}$ for some $\sigma \in \mathfrak S_n$.

Here, $\mathfrak S_n$ is the group of permutations of $[\![ 1, n ]\!]$,  and $x \simeq_H y$, for $x, y \in H$, means that $y \in H^\times x H^\times$.
Moreover, we define, for every $x \in H$, 
$$
\mathscr{Z}_H(x) := \pi_H^{-1}(x) \subseteq \mathscr{F}^\ast(\mathcal A(H))
$$ 
(the *set of factorizations* of $x$) and 
$$\mathsf L_H(x) := \{\|\mathfrak a\|_H: \mathfrak a \in \mathscr{Z}_H(x)\}$$
(the *set of lengths* of $x$). We call $H$ *atomic* if $\mathsf L_H(x) \ne \emptyset$ for all $x \in H \setminus H^\times$.

On a related note, let $H$ and $K$ be multiplicatively written monoids, and let $\varphi$ be a homomorphism $H \to K$. We write $\varphi^\ast$ for the unique homomorphism $\mathscr{F}^\ast(H) \to \mathscr{F}^\ast(K)$ such that $\varphi^\ast(x) = \varphi(x)$ for all $x \in H$, and we say that $\varphi$ is \textit{essentially surjective} if $K = K^\times \varphi(H) K^\times$ (this is actually an instance of the notion of essentially surjective functor in category theory), and an *equimorphism* (*from $H$ to $K$*) if the following hold:

 1. $\varphi(x) = 1_K$ for some $x \in H$ only if $x \in H^\times$, that is, $\varphi^{-1}(1_K) \subseteq H^\times$.
 2. $\varphi$ is *atom-preserving*, i.e., $\varphi(a) \in \mathcal A(K)$ for all $a \in \mathcal A(H)$.
 3. If $x \in H \setminus \{1_H\}$ and $\mathfrak b \in \mathscr{Z}_K(\varphi(x)) \ne \emptyset$, then $(\mathfrak b, \varphi^\ast(\mathfrak a)) \in \mathscr{C}_K$ for some $\mathfrak{a} \in \mathscr{Z}_H(x)$. 

Lastly, we say that $H$ is *essentially equimorphic* to $K$ if there exists an essentially surjective equimorphism from $H$ to $K$. I hope to haven't forgot anything.