I know why the caged bird sings.

aviary.m

// cc aviary.m -o aviary -framework Foundation -fobjc-arc && ./aviary
#import <Foundation/Foundation.h>

// use it like lambda(…args…)(…return value…)
#define lambda(...) \
^ (__VA_ARGS__) _lambda_body

#define _lambda_body(...) \
{ return __VA_ARGS__; }

/// I-Combinator
typeof(id(^)(id)) I() {
	return lambda(id x)(x);
}

/// K-Combinator
typeof(id(^)(id)) K(id x) {
	return lambda(id y)(x);
}

/// S-Combinator
typeof(typeof(id(^)(id))(^)(id)) S(typeof(id(^)(id))(^x)(id)) {
	return lambda(id(^y)(id))(lambda(id z)(x(z)(y(z))));
}

/// Y-Combinator
typeof(typeof(id(^)(id))(^)(id)) Y(id(^z)(id)) {
	return lambda(typeof(id(^)(id))(^f)(id))(lambda(id(^x)(id))((id(^)(id))f(x(x))))(lambda(id(^t)(id))(z));
}


/// Idiotbird Combinator; Alias for the I-Combinator.
typeof(id(^)(id))(*Idiotbird)() = I;

/// Kestral Combinator; Alias for the K-Combinator.
typeof(id(^)(id))(*Kestral)(id x) = K;

/// Starling Combinator; Alias for the S-Combinator.
typeof(typeof(id(^)(id))(^)(id))(*Starling)(typeof(id(^)(id))(^x)(id)) = S;



typeof(typeof(id(^)(id))(^)(id)) Bluebird(id(^bc)(id)) {
	return lambda(id(^ab)(id))(lambda(id a)(bc(ab(a))));
}

typeof(typeof(typeof(id(^)(id))(^)(id))(^)(typeof(id(^)(id, id)))) Blackbird(id(^cd)(id)) {
	return lambda(id(^abc)(id, id))(lambda(id a)(lambda(id b)(cd(abc(a, b)))));
}

typeof(typeof(typeof(typeof(id(^)(typeof(id(^)(id))))(^)(typeof(id(^)(id))))(^)(typeof(id(^)(id))))(^)(id)) Bunting(id(^de)(id)) {
	return lambda(id(^cd)(id))(lambda(id(^bc)(id))(lambda(id(^ab)(id))(lambda(id a)(de(cd(bc(ab(a))))))));
}

typeof(typeof(typeof(id(^)(typeof(id(^)(id))))(^)(typeof(id(^)(id))))(^)(id)) Becard(id(^cd)(id)) {
	return lambda(id(^bc)(id))(lambda(id(^ab)(id))(lambda(id a)(cd(bc(ab(a))))));
}

typeof(typeof(id(^)(id))(^)(id)) Cardinal(id(^abc)(id, id)) {
	return lambda(id a)(lambda(id b)(abc(b, a)));
}

typeof(typeof(typeof(id(^)(typeof(id(^)(id))))(^)(typeof(id(^)(id))))(^)(id)) Dove(id(^acd)(id, id)) {
	return lambda(id a)(lambda(id(^bc)(id))(lambda(id b)(acd(a, bc(b)))));
}


typeof(typeof(typeof(typeof(id(^)(id))(^)(typeof(id(^)(id))))(^)(id))(^)(id)) Dickcissel(id(^abde)(id, id, id)) {
	return lambda(id a)(lambda(id b)(lambda(id(^cd)(id))(lambda(id c)(abde(a, b, cd(c))))));
}

typeof(typeof(typeof(typeof(id(^)(id))(^)(typeof(id(^)(id))))(^)(id))(^)(typeof(id(^)(id)))) Dovekie(id(^cde)(id, id)) {
	return lambda(id(^ac)(id))(lambda(id a)(lambda(id(^bd)(id))(lambda(id b)(cde(ac(a), bd(b))))));
}

typeof(typeof(typeof(typeof(id(^)(id))(^)(typeof(id(^)(id, id))))(^)(id))(^)(id)) Eagle(id(^ade)(id, id)) {
	return lambda(id a)(lambda(id(^bcd)(id, id))(lambda(id b)(lambda(id c)(ade(a, bcd(b, c))))));
}

/// #Murica #Freedom #GunsAndBeer #BeerAndGuns
typeof(typeof(typeof(typeof(typeof(typeof(id(^)(id))(^)(id))(^)(typeof(id(^)(id, id))))(^)(id))(^)(id))(^)(typeof(id(^)(id, id)))) BaldEagle(id(^efg)(id, id)) {
	return lambda(id(^abe)(id, id))(lambda(id a)(lambda(id b)(lambda(id(^cdf)(id, id))(lambda(id c)(lambda(id d)(efg(abe(a, b), cdf(c, d))))))));
}

typeof(typeof(typeof(id(^)(typeof(id(^)(id, id))))(^)(id))(^)(id)) Finch(id a) {
	return lambda(id b)(lambda(id b)(lambda(id(^bac)(id, id))(bac(b, a))));
}

/// Joy-Combinator
typeof(typeof(id(^)(id))(^)(id)) Jalt(id(^ac)(id)) {
	return lambda(id a)(lambda(id b)(ac(a)));
}

typeof(id(^)(id)) Kite(id a) {
	return lambda(id b)(b);
}


/// O-Combinator
typeof(id(^)(id)) Owl(id(^aba)(id(^)(id))) {
	return lambda(id(^ab)(id))(aba(ab));
}