People often mention that there is an isomorphic nature between language and the world in the Tractatus' conception of language. As far as I can see it, according to Wittgenstein (it's been a few months since I actually read through it, so apologies if the terminology is wrong), the objects in the world correspond to the elementary names of the language. Thus, a sentence (a complex of elementary names) is a picture for a possible state of affairs. I guess the isomorphism is mentioned because in the ideal version of language Wittgenstein is talking about, there is a bijective mapping between the basic objects and the names for these objects. I imagine the relation that is preserved through the isomorphism is roughly: The relationship between the objects in the world is mirrored by the formal relationship between the names in a proposition.
So maybe in a simple natural language version of Wittgenstein, a statement like "the ball is 10m from the door", transformed through the isomorphism, corresponds to the state of affairs where this is the case.
I'm assuming that Wittgenstein must be talking about a radically new form of language here, because as it stands in natural language is more of a homomorphism than an isomorphism. That is to say, a statement like "The ball is 10cm from the door" corresponds to many different states of affairs (the ball could be 10cm north, or 10cm south, etc). So there is not a one-to-one mapping, as multiple states of affairs all get mapped to the same sentence. Given the common mention of isomorphism in relation to the picture theory, does that mean that, in Wittgenstein's view, when he is staking about propositions, he is imagining a type of language where literally every different state of affairs has its own isomorphic description, so that there would be an infinite number of propositions which would collectively make up the natural language idea of the ball being 10cm from the door.
If this is what Wittgenstein is talking about, then I guess it's an isomorphism, not a homomorphism, although in reality people's descriptive sentences could be better viewed as involving a homomorphism from reality to language. I guess Wittgenstein is talking about some language where even the simplest notion (the ball being 10cm away) would be an extraordinarily long proposition formally, at least as complicated formally as the world is.
Also, I am more of a fan of late-Wittgenstein, so I'm aware of how he came to the view that the work was fundamentally flawed, but I'm trying to get an idea of how it was supposedly going to work.