# Is there such a thing as $coW[1]$-hardness?

I have a problem $$\mathsf{A}$$ and I would like to analyze its (parameterized) computational complexity.

I found a parameterized reduction from the complement of the independent set ($$\mathsf{coIS}$$) problem (i.e. given a graph $$G$$ and an integer $$k$$, is there no independent set of size $$k$$ in G?), such that there is a parameter $$\kappa$$ to my new problem $$\mathsf{A}$$ that only increases by a constant in regards to $$k$$.

Formally, I have found a reduction function $$R$$, that takes as input an instance $$(G, k)$$ of $$\mathsf{coIS}$$, where $$G$$ is an arbitrary undirected graph and $$k \in \mathbf{N}$$ is some integer, and outputs an instance $$(I, \kappa)$$ of problem $$\mathsf{A}$$, such that:

1. $$(G, k) \in \mathsf{coIS} \Leftrightarrow (I, \kappa) \in \mathsf{A}$$ (alternatively: $$(G, k) \not\in \mathsf{IS} \Leftrightarrow (I, \kappa) \in \mathsf{A}$$).
2. $$R$$ runs in $$f(k) \cdot |G|^{\mathcal{O}(1)}$$ time, where $$f$$ is some computable function (in our case, $$f$$ is polynomial).
3. $$\kappa \leq h(k)$$, where $$h$$ is some computable function.

From this, I infer that problem $$\mathsf{A}$$ is coNP-hard. However, I would also like to make a statement about the parameterized complexity of problem $$\mathsf{A}$$. My intuition tells me that problem $$\mathsf{A}$$ is $$coW[1]$$-hard when parameterized by $$\kappa$$.

However, I have not been able to find any reference to the class $$coW[1]$$ in the literature or in any previously asked question. Given that my understanding of the class $$W[1]$$ and the $$W$$-hierarchy is somewhat primitive, I don't know if the notion of $$coW[1]$$ makes any sense. I could well imagine that $$W[1] = coW[1]$$ holds, but I don't know how one would prove such a statement.