Translation:The dog is sitting and the cat is lying.
Agree with Chiara that lying down should be accepted. If I say he is lying, what do I actually mean? its hard to tell unless you understand how people tend to use the words in practice (or at least how I've used the words in the UK to distinguish the meaning). Lying on its own is usually means dishonesty and to indicate position, we usually add more information like how or where someone is lying. So if you said he is lying, my default interpretation is he is being dishonest. If you say he is lying on the sofa, I know immediately you are talking about him lying down. Positional use usually adds a "down", "on", "in" i.e. I am lying down, he is lying on the floor, he is lying in bed. While its clear here that its a positional meaning as cats cant talk, as Chiara says, we just dont say lying when talking about body position because it becomes too confusing on the meaning. We always add a how or why bit to it to distinguish it from the being dishonest meaning so it just sounds very weird to read that!!
The problem is that to lie and to lay mean different things. You lay things on a table, you lie on a sofa. The complication, and where a lot of confusion arises even with native English speakers is the past tense of lie is lay.... and probably why it started being used interchangeably with lie at some point. For example, I lie on the sofa or I am lying on the sofa (present) but I lay on the sofa watching TV yesterday (past).