How well are Cuba's LGBT support programs actually implemented?

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P.S. The Br*tish have sent in a silly letter. We shall neither print it nor reply to them.

What's the waiting time and quality of care on this?

I don't know the answer to this but I do know that they have been temporarily cancelled during some crises, including the initial portions of the pandemic. Cuba is a poor country and the poverty forced on them has led to frequent "can we afford to address that?" discourse on what we would call social issues. Therefore, it would not surprise me if wait times were significant, though again I really don't know.

Is it actually well implemented or is it just a law with no funding behind it?

It is actually implemented and usually funded. Cubans regularly receive gender-affirming care including surgeries.

How tolerant is the Cuban culture?

Cubans have diversity of thought just like everywhere else. It has substantial queer rights movements driving these things forward and pushback from conservatives. These policies were achieved democratically, via participatory democracy, and you can be certain they're popular enough compared to many other places (e.g. the UK banning puberty blockers for minors).

It's definitely worthwhile to continue looking into the various attitudes towards trans people in Cuba. I'm by no means an expert.

Is the remaining intolerance mostly from the Spanish colony's Catholicism or did Castro's homophobia leave a lasting impact?

Castro publicly rejected homophobia later on. I do know that homophobia in Cuba is not exclusive to Catholicism, at least not explicitly, and that atheist communists there can still be homophobic. It's possible to argue that they inherited cultural baggage from catholicism, but many communists have also historically found ways to rationalize conservative opinions as correct communist ideas that fly in the face of liberalism. This is easier for jerks like that to discover nowadays because liberals pay lip service to queer rights. I can't give you a sense of sense of how common these different justifications for homophobia are because I don't know that myself, but hopefully these tidbits are useful.

Why was Castro homophobic?

I'm not sure whether we have much information on Castro's explicitly stated personal views. Che wrote about this and his regretting being homophobic, as he also later rejected homophobia. Castro has mainly associated the policies of sending gay people to work camps as a response to external attack by the US and the fact that gay people, namely gay men, were not allowed in the military and therefore free from mandatory service, but the state still felt the onus to force work on them, hence sending them to work camps. Keep in mind this is what was said long after the fact by a very old Fidel, it would be wise to do more research on this.

Were there really concentration camps?

Sort of? They were work camps and specific groups were sent there to remove them from society - hardcore Catholics, gay people, etc - so I think that likely fits the definition. But unlike how concentration camps were used by settlers and fascists, these were temporary and more like POW camps than the conditions forced on Africans by the British or Jewish people et al in Nazi campa. Not that "just" putting gay people in a less harsh forced labor camp is justified, but it is very different from the way the OG concentration camps were run, and for what purpose.

In a country which incarcerated LGBT people for most of Fidel Castro’s rule, the island’s capital now has gay clubs and bars and celebrates Pride every year.

As if The UK isn't just straight up on the edge of incarcerating Trans folks for existing and wanting healthcare.